Since it says SHOULD, it is a strong
suggestion that it be done immediately (and at my table it is), but it is not
an imperative (Not a YOU MUST).

So, based on Andy’s ruling, as long
as you draw and play all the required cards, before the end of your turn, you
can do so in what ever order you like.

The rule also says “IF NEEDED”,
this is to rule out drawing more cards then necessary, I.E. playing a draw 5,
drawing the 5 cards and then playing a draw 3. You only get the original 5
cards, not 8 cards.

Hopefully this has put to rest any
question on when to draw cards.

And remember, if you feel strongly that
you MUST draw first, just make it a house rule.